Exceptional collections of four line bundles on Beauville's quotient-product surface

Beauville constructed surface of general type with $p_g = 0$, $K^2 = 8$ and hence Euler number four. We show that exactly 39 line bundles on this surface are acyclic. Moreover, up to twist there are 6 exceptional collections of four line bundles. An orthogonal to any such collection is yet another example of quasi-phantom — non-trivial category with vanishing Hochschild homology. This is joint work with Evgeny Shinder.